For example, it turns out that we need an identity that will change the form of sin

^{2}x, in order to evaluate the integral of sin

^{2}x. While my students were taking a test on volumes of rotation a few weeks back, I was thinking about integration techniques, which I’d be teaching next. I wanted to solve this problem, and had no table of trig identities handy. So I thought I’d draw a graph to see what it might tell me.

The next day of class I showed it to my students, and today a student who was absent asked to see my 'derivation'. I was tickled. And then I thought maybe some of you might like to see it. This is how I

*can*remember things - visually. Because then everything is connected.

I want to take each y-coordinate on this graph of y=sin x, and square it, to get y= sin

^{2}x. The points where y=0 stay put, and so does the point where y=1. The point at (3π/2,-1) moves to (3π/2,1). The portions of the graph very close to y=0 look almost like straight lines, so squaring those portions will get us a shape that’s close to a parabola near the y=0 points.

Now we have something that looks like the new graph I've added in here. Well, that looks like a trig function - what would it be? The frequency has doubled, so we use 2x. It looks like an upside down cosine, with horizontal midline at y= 1/2, and amplitude (distance from midline to top or bottom) = 1/2. So I get ...

y = 1/2 - 1/2 * cos(2x), which is usually written as ...

y = (1-cos 2x)/2. (Apologies for my limited html skills.)

Now that you've seen this, can you find the identity for cos

^{2}x?

"Now that you've seen this, can you find the identity for cos^2 x?"

ReplyDeleteCan I use sin^2 x + cos^2 x = 1 along with sin^2 x = (1 - cos 2x)/2? :)

Sure. But I think graphing directly is almost easier than the algebra of this. Well, maybe not... but close. And cooler. ;^)

ReplyDeleteI think algebra often hides relationships that other approaches keep visible.

The "straight line -> parabola" part seems to be hard for kids I've worked with on this one. (They have a lot of transformation experience so the scaling and shifting comes easily.)

ReplyDeleteI've also noticed that if you're not careful, people draw the humps of the graph as semicircles in which case they are like vertical lines instead of like y=x where they cross the axis. I make a much bigger deal about this now and I hope it pays off when they get to calculus classes later.

Thanks, Josh. This is the first time I've shown this to students at all. I think it's kind of new for me to think this way. I'll try to watch how they draw their sine waves.

ReplyDelete